tppreqb - Metamath Proof Explorer

	Metamath Proof Explorer	< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tppreqb		Structured version Unicode version

Theorem tppreqb 4079

Description: An unordered triple is an unordered pair if and only if one of its elements is a proper class or is identical with one of the another elements. (Contributed by Alexander van der Vekens, 15-Jan-2018.)

**Assertion**
Ref	Expression
tppreqb	$/\$ $/\$

**Proof of Theorem tppreqb**
Step	Hyp	Ref	Expression
1		3ianor 999	. . . 4 $/\$ $/\$ $\/$ $\/$
2		df-3or 983	. . . 4 $\/$ $\/$ $\/$ $\/$
3	1, 2	bitri 252	. . 3 $/\$ $/\$ $\/$ $\/$
4		orass 526	. . . . 5 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
5		ianor 490	. . . . . . . 8 $/\$ $\/$
6		tpprceq3 4078	. . . . . . . 8 $/\$
7	5, 6	sylbir 216	. . . . . . 7 $\/$
8		tpcoma 4034	. . . . . . 7
9		prcom 4016	. . . . . . 7
10	7, 8, 9	3eqtr3g 2480	. . . . . 6 $\/$
11		orcom 388	. . . . . . . 8 $\/$ $\/$
12		ianor 490	. . . . . . . 8 $/\$ $\/$
13	11, 12	bitr4i 255	. . . . . . 7 $\/$ $/\$
14		tpprceq3 4078	. . . . . . 7 $/\$
15	13, 14	sylbi 198	. . . . . 6 $\/$
16	10, 15	jaoi 380	. . . . 5 $\/$ $\/$ $\/$
17	4, 16	sylbi 198	. . . 4 $\/$ $\/$ $\/$
18	17	orcs 395	. . 3 $\/$ $\/$
19	3, 18	sylbi 198	. 2 $/\$ $/\$
20		df-tp 3941	. . . 4
21	20	eqeq1i 2428	. . 3
22		ssequn2 3577	. . . 4
23		snssg 4071	. . . . . . 7
24		elpri 3953	. . . . . . . 8 $\/$
25		nne 2600	. . . . . . . . . 10
26		3mix2 1175	. . . . . . . . . 10 $\/$ $\/$
27	25, 26	sylbir 216	. . . . . . . . 9 $\/$ $\/$
28		nne 2600	. . . . . . . . . 10
29		3mix3 1176	. . . . . . . . . 10 $\/$ $\/$
30	28, 29	sylbir 216	. . . . . . . . 9 $\/$ $\/$
31	27, 30	jaoi 380	. . . . . . . 8 $\/$ $\/$ $\/$
32	24, 31	syl 17	. . . . . . 7 $\/$ $\/$
33	23, 32	syl6bir 232	. . . . . 6 $\/$ $\/$
34		3mix1 1174	. . . . . . 7 $\/$ $\/$
35	34	a1d 26	. . . . . 6 $\/$ $\/$
36	33, 35	pm2.61i 167	. . . . 5 $\/$ $\/$
37	36, 1	sylibr 215	. . . 4 $/\$ $/\$
38	22, 37	sylbir 216	. . 3 $/\$ $/\$
39	21, 38	sylbi 198	. 2 $/\$ $/\$
40	19, 39	impbii 190	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 187 $\/$ wo 369 $/\$ wa 370 $\/$ w3o 981 $/\$ w3a 982

wceq 1437

wcel 1872

wne 2594

cvv 3017

cun 3372

wss 3374

csn 3936

cpr 3938

ctp 3940

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1663 ax-4 1676 ax-5 1752 ax-6 1798 ax-7 1843 ax-10 1891 ax-11 1896 ax-12 1909 ax-13 2058 ax-ext 2403

This theorem depends on definitions: df-bi 188 df-or 371 df-an 372 df-3or 983 df-3an 984 df-tru 1440 df-ex 1658 df-nf 1662 df-sb 1791 df-clab 2410 df-cleq 2416 df-clel 2419 df-nfc 2553 df-ne 2596 df-v 3019 df-dif 3377 df-un 3379 df-in 3381 df-ss 3388 df-nul 3700 df-sn 3937 df-pr 3939 df-tp 3941

This theorem is referenced by: (None)

W3C validator